Topological Defects in Liquid Crystal Films

A topological theory of liquid crystal films in the presence of defects is developed based on the Ф-mapping topological current theory. By generalizing the free-energy density in ＂one-constant＂ approximation, a covariant free- energy density is obtained, from which the U（1） gauge field and the unified topological current for monopoles and strings in liquid crystals are derived. The inner topological structure of these topological defects is characterized by the winding numbers of Ф-mapping.

A novel approach to topological defects in a vector order parameter system

Based on Duan＇s topological current theory,we propose a novel approach to study the topological properties of topological defects in a two-dimensional complex vector order parameter system.This method shows explicitly the fine topological structure of defects.The branch processes of defects in the vector order parameter system have also been investigated with this method.

Topological defects on solutions of the non-relativistic equation for extended double ring-shaped potential

In this study, we focus into the non-relativistic wave equation described by the Schrodinger equation, specifically considering angular-dependent potentials within the context of a topological defect background generated by a cosmic string. Our primary goal is to explore quasi-exactly solvable problems by introducing an extended ring-shaped potential. We utilize the Bethe ansatz method to determine the angular solutions, while the radial solutions are obtained using special functions. Our findings demonstrate that the eigenvalue solutions of quantum particles are intricately influenced by the presence of the topological defect of the cosmic string,resulting in significant modifications compared to those in a flat space background. The existence of the topological defect induces alterations in the energy spectra, disrupting degeneracy.Afterwards, we extend our analysis to study the same problem in the presence of a ring-shaped potential against the background of another topological defect geometry known as a point-like global monopole. Following a similar procedure, we obtain the eigenvalue solutions and analyze the results. Remarkably, we observe that the presence of a global monopole leads to a decrease in the energy levels compared to the flat space results. In both cases, we conduct a thorough numerical analysis to validate our findings.

Topological defects strengthened nonradical oxidation performance of biochar catalyzed peroxydisulfate system

Nonradical oxidation based on peroxydisulfate(PDS)activation has attracted increasing attention for selective degradation of organic pollutants.Herein,topological defects were introduced into biochar(BC)via removing N atoms in N-doped BC(NBC)in an attempt to improve the nonradical catalytic performance.Compared to the pristine BC and NBC,the introduction of topological defects could achieve up to 36.6-and 8.7-times catalytic activity enhancement,respectively.More importantly,it was found that the catalytic activity was dominated by topological defects,which was verified by the significant positive correlation between the pseudo-first-order rate constants and the content of topological defects.Theoretical calculations suggested that topological defects enhanced the electrondonating ability of BC by reducing the energy gap,which made the electrons transfer to PDS molecules more easily.As a result,holes were generated after the carbon defects lost electrons,and induced a nonradical oxidation process.Benefiting from the merits of nonradical oxidation,the developed BC/PDS system showed superior performance in removing electron-rich contaminants in the presence of inorganic anions and in the actual environments.This study not only provides a potential avenue for designing efficient biochar-based catalysts,but also advances the mechanism understanding of nonradical oxidation process induced by carbon defects.

“Charging”the cigarette butt:heteroatomic porous carbon nanosheets with edge-induced topological defects for enhanced oxygen evolution performance

Owing to the complexity of electron transfer pathways,the sluggish oxygen evolution reaction process is defined as the bottleneck for the practical application of Zn-air batteries.In this effort,metal nanoparticles(Co,Ni,Fe,etc.)encapsulated within nitrogen-doped carbon materials with abundant edge sites were synthesized by one-step pyrolysis treatment using cigarette butts as raw materials,which can drastically accelerate the overall rate of oxygen evolution reaction by facilitating the adsorption of oxygenated intermediates by the edge-induced topological defects.The prepared catalyst of nitrogen-doped carbon porous nanosheets loaded with Co nanoparticles(Co@NC-500)exhibits enhanced catalytic activity toward oxygen evolution reaction,with a low overpotential of 350 mV at the current density of 10 mA·cm^(-2).Furthermore,the Zn-air battery assembled with Co@NC-500 catalyst demonstrates a desirable performance affording an open-circuit potential of 1.336 V and power density of 33.6 mW·cm^(-2),indicating considerable practical application potential.

Visualizing the charge order and topological defects in an overdoped （Bi,Pb）2Sr2CuO6+x superconductor

Electronic charge order is a symmetry breaking state in high-Tc cuprate superconductors. In scanning tunneling microscopy, the detected charge-order-induced modulation is an electronic response of the charge order. For an overdoped(Bi,Pb)2Sr2CuO6+x sample, we apply scanning tunneling microscopy to explore local properties of the charge order. The ordering wavevector is nondispersive with energy, which can be confirmed and determined. By extracting its order-parameter field, we identify dislocations in the stripe structure of the electronic modulation, which correspond to topological defects with an integer winding number of ±1. Through differential conductance maps over a series of reduced energies, the development of different response of the charge order is observed and a spatial evolution of topological defects is detected. The intensity of charge-order-induced modulation increases with energy and reaches its maximum when approaching the pseudogap energy. In this evolution, the topological defects decrease in density and migrate in space. Furthermore, we observe appearance and disappearance of closely spaced pairs of defects as energy changes. Our experimental results could inspire further studies of the charge order in both high-Tccuprate superconductors and other charge density wave materials.

Topological hierarchy matters — topological matters with superlattices of defects

Topological insulators/superconductors are new states of quantum matter with metallic edge/surface states.In this paper,we review the defects effect in these topological states and study new types of topological matters — topological hierarchy matters.We find that both topological defects（quantized vortices） and non topological defects（vacancies） can induce topological mid-gap states in the topological hierarchy matters after considering the superlattice of defects.These topological mid-gap states have nontrivial topological properties,including the nonzero Chern number and the gapless edge states.Effective tight-binding models are obtained to describe the topological mid-gap states in the topological hierarchy matters.

Topological Vortices in a Spin-Triplet Superconductor

Based on the complex three-component order parameter model of a spin-triplet superconductor, by using the C-mapping theory, we derive a new equation describing the distribution of the magnetic field for vortices, which can be reduced to the modified London equation in the case of ｜ψ^2｜^2 ^- ｜ψ^3｜^2 = 0 and Wl^1= 1. A magnetic flux quantization condition for vortices in a spin-triplet superconductor is also derived, which is topological-invariant. Fhrthermore, the branch processes during the evolution of the vortices in a spin-triplet superconductor are discussed. We also point out that the sum of the magnetic flux quantization that those vortices carried is 2nФo （Фo is the unit magnetic flux）, that is to say, the sum of winding number is even, which needs to be proved by experiment.

Topological aspects in a two-component Bose condensed system in a neutron star

By making use of Duan-Ge＇s decomposition theory of gauge potential and the topological current theory proposed by Prof. Duan Yi-Shi, we study a two-component superfluid Bose condensed system, which is supposed to be realized in the interior of neutron stars in the form of the coexistence of a neutron superfluid and a protonic superconductor. We propose that this system possesses vortex lines. The topological charges of the vortex lines are characterized by the Hopf indices and the Brower degrees of C-mapping.

Topological Aspects of Triplet Superconductors

In this paper, using the Φ-mapping theory, it is shown that two kinds of topological defects, i.e., the vortex lines and the monopoles exist in the helical configuration of magnetic field in triplet superconductors. And the inner topological structure of these defects is studied. Because the knot solitons in the triplet superconductors are characterized by the Hopf invariant, we also establish a relationship between the Hopf invariant and the linking number of knots family, and reveal the inner topological structure of the Hopf invariant.

First principle study of edge topological defect-modulated electronic and magnetic properties in zigzag graphene nanoribbons

Zigzag graphene nanoribbon （ZGNR） is a promising candidate for next-generation spintronic devices. Development of the field requires potential systems with variable and adjustable electromagnetic properties. Here we show a detailed investigation of ZGNR decorated with edge topological defects （ED-ZGNR） synthesized in laboratory by Ruffieux in 2015 [Pascal Ruffieux, Shiyong Wang, Bo Yang, et al. 2015 Nature 531 489]. The pristine ED-ZGNR in the ground state is an antiferromagnetic semiconductor, and the acquired band structure is significantly changed compared with that of perfect ZGNR. After doping heteroatoms on the edge, the breaking of degeneration of band structure makes the doped ribbon a half-semi-metal, and nonzero magnetic moments are induced. Our results indicate the tunable electronic and magnetic properties of ZGNR by deriving unique edge state from topological defect, which opens a new route to practical nano devices based on ZGNR.

Ground-state phases and spin textures of spin–orbit-coupled dipolar Bose–Einstein condensates in a rotating toroidal trap

We investigate the ground-state phases and spin textures of spin-orbit-coupled dipolar pseudo-spin-1/2 Bose-Einstein condensates in a rotating two-dimensional toroidal potential.The combined effects of dipole-dipole interaction(DDI),spin-orbit coupling(SOC),rotation,and interatomic interactions on the ground-state structures and topological defects of the system are analyzed systematically.For fixed SOC strength and rotation frequency,we provide a set of phase diagrams as a function of the DDI strength and the ratio between inter-and intra-species interactions.The system can show rich quantum phases including a half-quantum vortex,symmetrical(asymmetrical)phase with quantum droplets(QDs),asymmetrical segregated phase with hidden vortices(ASH phase),annular condensates with giant vortices,triangular(square)vortex lattice with QDs,and criss-cross vortex string lattice,depending on the competition between DDI and contact interaction.For given DDI strength and rotation frequency,the increase of the SOC strength leads to a structural phase transition from an ASH phase to a tetragonal vortex lattice then to a pentagonal vortex lattice and finally to a vortex necklace,which is also demonstrated by the momentum distributions.Without rotation,the interplay of DDI and SOC may result in the formation of a unique trumpet-shaped Bloch domain wall.In addition,the rotation effect is discussed.Furthermore,the system supports exotic topological excitations,such as a half-skyrmion(meron)string,triangular skyrmion lattice,skyrmion-halfskyrmion lattice,skyrmion-meron cluster,skyrmion-meron layered necklace,skyrmion-giant-skyrmion necklace lattice,and half-skyrmion-half-antiskyrmion necklace.

A New Approach to the Dark Matter/Dark Energy Puzzle

The dilemmas posed by dark matter and dark energy have been with us for decades without a satisfactory resolution. We propose that both DM and DE can be explained by the existence of long-lived topological gravitational vortices that were produced in the quark-gluon epoch of cosmic inflation due to the misalignment of the gravitational and strong forces. This is analogous to the misalignment mechanism proposed for the production of axions in the early universe. The masses of these topological vortices are expected to be on the order of the nucleon mass. Possible means for their detection are discussed.

Formation and Evolution of Wakes in the Spacetime Generated by a Cosmic String in <i>f</i>(<i>R</i>) Theory of Gravity

The formation and evolution of cosmic string wakes in the framework of a f(R) theory of gravity are investigated in this work. We consider a simple model in which baryonic matter flows past a cosmic string. We treat this problem in the Zel’dovich approximation. We compare our results with previous results obtained in the context of General Relativity and Scalar-Theories of Gravity.

Stability of Disclinations in Nematic Liquid Crystals

In the light of C-mapping method and topological current theory, the stability of disclinations around a spherical particle in nematic liquid crystals is studied. We consider two different defect structures around a spherical particle： disclination ring and point defect at the north or south pole of the particle. We calculate the free energy of these different defects in the elastic theory. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclinations. The other is the free energy density of disclinations themselves, which is shown to be concentrated at the defect and to be topologically quantized in the unit of （k - k24）π/2. It is shown that in the presence of saddle-splay elasticity a dipole （radial and hyperbolic hedgehog） configuration that accompanies a particle with strong homeotropic anchoring takes the structure of a small disclination ring, not a point defect.

Quench dynamics in 1D model with 3rd-nearest-neighbor hoppings

The non-equilibrium dynamics of a one-dimensional(1 D)topological system with 3 rd-nearest-neighbor hopping has been investigated by analytical and numerical methods.An analytical form of topological defect density under the periodic boundary conditions(PBC)is obtained by using the Landau-Zener formula(LZF),which is consistent with the scaling of defect production provided by the Kibble-Zurek mechanism(KZM).Under the open boundary conditions(OBC),quench dynamics becomes more complicated due to edge states.The behaviors of the system quenching across different phases show that defect production no longer satisfies the KZM paradigm since complicated couplings exist under OBC.Some new dynamical features are revealed.

Branch Processes of Regular Magnetic Monopole

In this paper, by making use of Duan＇s topological current theory, the branch process of regular magnetic monopoles is discussed in detail. Regular magnetic monopoles are found generating or annihilating at the limit point and encountering, splitting, or merging at the bifurcation point and the degenerate point systematically of the vector order parameter field φ（x）. Furthermore, it is also shown that when regular magnetic monopoles split or merge at the degenerate point of field function φ, the total topological charges of the regular magnetic monopoles are stilI unchanged.

Effect of Saddle-Splay Elasticity on Stability of Disclination Rings in Nematic Liquid Crystals

In this paper, the stability of disclination ring in nematie liquid crystals is studied. In the presence of saddle-splay elasticity （characterized by k24） the disclination ring has a universal equilibrium radius. Depending on the values of the saddle-splay constant k24, the universal equilibrium radius is altered. When k24 〉 0.92k （m = 1/2） and k24 〉 0.88k （m =-1/2）, the disclination will be a point rather than a ring, where k is the Frank elastic constant in the one-constant approximation.

Lamb wave topological imaging for blind hole defects in isotropic plates

The research aims at validating the ability of topological imaging to blind holes in isotropic plates using Lamb waves. Due to the defect is not symmetric around the mid- plane of the plate, the effect of Lamb mode conversion will have to be taken into account. The imaging method is based on two computations of ultrasonic fields, one forward and one adjoint, performed for the defect-free reference medium. The excited signal and scattered Lamb waves caused by the blind hole, are used as emitting sources to compute the forward problem and the adjoint problem, respectively. With the help of the finite element simulations, the natural refocusing process of the multimode Lamb waves at the defect location is visually demonstrated by the transient acoustic field snapshots at the different moments to strengthen the physical mechanism of the topological imaging method. The numerical results demonstrate that topological imaging has relatively stronger applicability to the blind hole in contrast to classical Delay And Sum （DAS） method and Time Reversal （TR） method. The topological imaging could handle complex Lamb wave signals containing mode conversions without the imaging quality being affected. The proposed imaging method presents a certain developing potential for detecting and imaging asymmetric defects in plate-like configurations using Lamb waves.

Boosting oxygen reduction in acidic media through integration of Pt-Co alloy effect and strong interaction with carbon defects

Optimization of Pt atom utilization efficiency is critical for the development of proton-exchange-membrane fuel cells.Here we aim to develop an efficient oxygen reduction reaction(ORR)catalyst with a low Pt content through the concurrent modification of Pt-Co alloy catalysts and carbon substrate.In the present study,ultrafine Pt-Co alloy nanoparticles are successfully synthesized and stabilized by topological carbon defects via adopting the ammonia thermal treatment.Despite the low Pt loading,the obtained catalyst exhibits an impressive half-wave potential of 0.926 V versus the reversible hydrogen electrode in 0.1 M HClO_(4)electrolyte.Furthermore,the durability testing using the timed-current method demonstrates a tiny loss of only 3.6%after 12 h.Both experimental results and theoretical calculations demonstrate that topological carbon defects significantly enhance the charge transfer processes at the alloy/carbon interface,contributing to the strong electronic metal-support interactions between the Pt-Co alloy nanoparticles and topological carbon defects.These interactions,along with the alloy effect,play a crucial role in promoting the ORR performance in acidic media.